Any day in the year can be uniquely
identified with two numbers. For whatever reason I want the number
with the smaller range to go first, so let's have an order of
MonthDay. The Month number ranging from 1 to 12 and the day number
ranging from 1 to 31. In an ideal world that would be all of the
information you needed, but there are not 372 days in the year so
some of the identifications people might make with those rules, say,
“231,” are not real days.
But say I didn't care about that, say I
didn't care about the number of days in the year at all and I just
wanted to make up a calendar. Say I wanted it to be 210 days. 210
is seven times thirty, so I should be able to make a calendar that
looks like XYY where X goes from 1 to 7, and YY goes from 01 to 30.
I could make one, so could you.
How would you do it?
I'm guessing that most people reading
would create seven months of 30 days each. So you'd have 101, 102,
103, all the way up to 130, which would be followed by 201, 202,
and so on. Some people might decide to have 30 weeks instead. So it
would be 101, 201, … , 601, 701, 102, 202 and so on.
I'm guessing that because our calendar
is built by dividing things up at different scales. We divide the
year into 12 months, then divide each of the months into days, and
each of the days into hours, and so on. Switching from one cycle to
another is like zooming in or zooming out. We almost never have two
different things going on at the same scale.
The Long Count I discussed in the last
post is also built on that kind of thinking with each unit divided up
by a unit below it, used to divide up a unit above it, or both.
Never were there two things happening side by side on the same scale.
As I said, I don't really find that interesting.
The Mayan calendars that I do find
interesting work entirely differently.
If you wanted to make the calendar I
ordered above in the style of one of those Mayan calendars you
wouldn't divide and then divide the divisions. You'd simply start
counting seven day weeks and thirty day months. You wouldn't have
one number stay fixed while the other cycled through, you'd have both
numbers moving at the same time. 101 would be the first day, 202
the second. After 707 would come 108. The thirtieth day would be
230, and the thirtyfirst 301.
Each day still has a unique identifier,
you've still got 210 days mapped out so that you can find any day in
it by being told it's XYY, but the way we arrived there is
completely different. Instead of naming months or numbering weeks or
something like that, you've just got months and weeks and are letting
them work alongside each other to create a larger system than either
makes on its own.
In this system staying something
happened on Friday the 13^{th} narrows it down to one day
every seven months, because Friday the 13^{th} comes on a
regular schedule. Which means that if 210 days is the period you're
concerned with, you don't have to name the months. And if you did
want to name them, you could just name them after the day they start
on since each month in that period starts on a different day.
We could also imagine a system where
the months were exactly the same as we're used to, but the there was
no leap year (Or if leap day didn't count as a day of the week that
would work too.) In that case, weeks would fit together with years
such that, say, Sunday the first of January only came once every
seven years. It would actually fit together quite nicely because on
the next year the first of January would be Monday, then Tuesday the
year after and so on. The result would be a seven year cycle. Years
within the cycle wouldn't need to be numbered because the year that
starts on Wednesday clearly comes two years after the one that starts
on Monday.
These probably aren't the best
examples, in part because I'm trying to use intervals we're familiar
with, a seven day week, a thirty day month, a 365 day year.
That said, I think they're at least
somewhat instructive. They show how you could set things up, they
also, I think, how confusing this whole thing seems. I don't know
about anyone else, but I definitely can't quickly figure out if two
dates are close together or far apart. It would be pretty
straightforward to know what day tomorrow or yesterday is. But if
you pick two random dates in the month and week calendar I described,
say Monday the 1^{st} and Friday the 13^{th}, it
isn't immediately clear to me if they're very close together or very
far apart. (I had to derive a formula for converting between a
weekmonth calendar and one where the days were numbered 1 thorough
210 to figure it out.)
On the other hand if I say the 1^{st}
of month 5 and the 13^{th} of month 1, you've got a much
better idea of how far apart they are (though do remember that there
are seven months here, not twelve.) So there's definitely something
to be said for the way we do things. And when I get to the Maya's
actual calendars you'll see that they seem to have realized that.
Unfortunately it doesn't look like I'll be getting to it today.
So I want to close this with some other
hypothetical calendars just to illustrate some points.
Seven and thirty make a calendar with
7*30 days because they don't have any factors in common. If I'd
chosen six and thirty it wouldn't have worked at all. The
thirtieth day would have been 630, and the thirtyfirst day would
have been 101, right back at the beginning. The six day cycle would
function as a subdivision of the 30 day month and nothing more. This
is, obviously, because six divides 30. There's no remainder so the
six cycle is back to the beginning after 30 days.
If I'd decided that I wanted to get
closer to the number of months in a year and chosen 12 instead of
seven, that wouldn't have worked quite right either. It would have
been a perfectly legitimate calendar, but it would have been a much
shorter one. It would have been 60 days instead of the 360 you might
expect if you thought there would be 12*30 days. The thirtieth day
would have been 0630, which means that in another thirty days you'd
find yourself at 1230, and the day after would be 0101.
I haven't mentioned this, but it's
probably worth noting that you can feel completely free to add dates.
If X days into a calendar is AC, and Y days in is BD then X+Y days
in is (A+B)(C+D). And now I'm regretting using a dash, which looks
an awful lot like a minus symbol, to separate the numbers in the
dates. Anyway, that works. The only thing is that the number you
get might be too high. As in the last example if you add 0630 to
itself you get 1260, but you can't have a number higher than 30 in
the second slot. So you just subtract a 30 and you get 1230, which
is the actual date you're looking for. (If the number in the first
slot had been to high you'd subtract a 12.)
Anyway, digression over, there is in
fact a rule for how many days are in cycle created by using smaller
cycles. There are as many as the least common multiple of the
smaller cycles. The least common multiple of two or more numbers is
just the smallest positive number (zero is not a positive number)
that is a multiple of all of the numbers in question. 210 is the
smallest positive number that is a multiple of both seven and thirty.
60 is the smallest positive number that is a multiple of both twelve
and thirty. 30 is the smallest positive number that is a multiple of
both six and thirty.
So if we decided to expand our 210 day
calendar by adding a 12 cycle to go alongside the 7 and 30 cycles,
the result would be a 420 day calendar, as that is the least common
multiple of 7, 12, and 30.
So, right now I'm going to sleep but
next time I plan to get to actual Mayan Calendars that use this sort
of system.

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